Class 10 Maths Chapter 7 Exercise 7.4 Solutions 2026 – Trigonometry Notes PDF

Unit 7: Trigonometry | Exercise 7.4 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 7.4 solutions for Trigonometry have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.

Trigonometry Solving Real Problems

The final exercise of this chapter takes the ratios and identities learned so far out of the abstract and into real-world measurement problems — finding the height of a building, the distance across a river, or how far a ladder needs to be from a wall, all without measuring them directly.

Key Concepts Covered in Exercise 7.4

Angle of Elevation

The angle of elevation is the angle measured upward from the horizontal to a point above, such as looking up at the top of a tower. It is used together with a known distance to find an unknown height.

Angle of Depression

The angle of depression is the angle measured downward from the horizontal to a point below, such as looking down from a cliff at a boat. By alternate angles, the angle of depression from one point equals the angle of elevation from the other.

Setting Up the Right Triangle

Every word problem in this exercise starts with translating the situation into a labeled right triangle — identifying the known angle, the known side, and which side is being asked for — before choosing the correct trigonometric ratio.

Choosing the Correct Ratio

Once the triangle is labeled, the choice between sin, cos, or tan depends on which two sides (opposite, adjacent, hypotenuse) are involved in the specific problem.

Step-by-Step Solved Examples

Q. No. 1: A tower casts a shadow 20 m long when the angle of elevation of the sun is 30°. Find the height of the tower.

Let height = h. tan30° = opposite/adjacent = h/20

h = 20 × tan30° = 20 × (1/√3)

h = 20/√3 ≈ 11.5 m

Q. No. 2: From the top of a cliff 50 m high, the angle of depression of a boat is 45°. Find the distance of the boat from the base of the cliff.

The angle of depression (45°) equals the angle of elevation from the boat (alternate angles).

tan45° = opposite/adjacent = 50/d

Since tan45° = 1: d = 50 m

Q. No. 3: A ladder 10 m long leans against a wall, making a 60° angle with the ground. Find how high up the wall the ladder reaches.

Let height = h. sin60° = opposite/hypotenuse = h/10

h = 10 × sin60° = 10 × (√3/2)

h = 5√3 ≈ 8.7 m

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 7.4 in the same numbered, step-by-step format.

MCQs, Short Questions & Long Questions from This Exercise

Alongside the main exercise, this page’s uploaded notes are organized the way Punjab Board papers are structured, so students can practice by question type:

• MCQs: Quick questions on identifying whether a scenario describes elevation or depression
• Short Questions: Simple height or distance problems using one standard angle, like Q. No. 1 above
• Long Questions: Multi-step problems, sometimes involving two angles or two right triangles together

Where This Matters Beyond This Exercise

This exercise is the practical payoff for the entire chapter, showing exactly why angle measurement, the six ratios, and the identities were worth learning. These same techniques are used directly in surveying, architecture, navigation, and astronomy, wherever a height or distance must be found indirectly using an angle and a known length.

Common Mistakes Students Make in Exercise 7.4

• Drawing the right triangle incorrectly, misplacing the angle of elevation or depression
• Choosing the wrong ratio for the given sides — for example, using sin instead of tan when the hypotenuse isn’t involved
• Forgetting that the angle of depression equals the angle of elevation from the other point (alternate angles)
• Leaving an answer as an unsimplified radical or fraction, such as 20/√3 without rationalizing

Why This Exercise Matters for the Board Exam

Word problems using angles of elevation and depression are among the most common long-question types in the trigonometry section, since they combine diagram-drawing, ratio selection, and calculation in one question. A clearly labeled diagram is often specifically rewarded with its own mark, even before any calculation begins.

Quick Links – Chapter 7: Trigonometry

Links for sections other than Exercise 7.4 will be activated as they are published on this site.

Download Exercise 7.4 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 7.4 notes free to download?

Yes, all notes on this page are completely free to view and download in PDF format.

Q2. Which board are these notes for?

These notes are prepared according to the Punjab Board syllabus and are useful for all Punjab boards (Lahore, Gujranwala, Multan, Sargodha, Rawalpindi, Faisalabad, DG Khan, Bahawalpur, Sahiwal) as well as the Federal Board (FBISE).

Q3. What is the difference between angle of elevation and angle of depression?

The angle of elevation is measured upward from the horizontal to an object above the observer, while the angle of depression is measured downward from the horizontal to an object below. By alternate angles, they are equal when measured between the same two points from opposite ends.

Q4. Does this page include MCQs and short questions too?

Yes, the uploaded notes include MCQs, short questions, and long questions related to this exercise’s topics, organized by question type to match the board paper pattern.

Q5. How can I download the PDF?

Click the “Download PDF” button above and the notes will open or download directly to your device.

Q6. Are these notes updated for the new 2026–27 syllabus?

Yes, these notes are prepared strictly according to the latest PCTB/PECTAA syllabus for the 2026–27 academic session.

Comments & Feedback

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